trig problem. please explain, is the answer not 9.76 hours?

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 75 and 85 degrees during the day and the average daily temperature first occurs at 9 AM. How many hours after midnight, to two decimal places, does the temperature first reach 79 degrees?

Answer 1

Let T = Asin(Bt + C) + D,

where T is temperatura in F, t is time in hours.

Then D = (75 + 85)/2 = 80°F, A = (85 - 75°)/2 = 5°F.

We have now equation: T = 5sin(At + C) + 80.

Letcoount timeromfmidnite. Then if t = 9 then T = 80°F. We get equation: 8i0 = 5sin(9B + C) + 80,

5sin(9B + C) = 0, sin (9B + C ) = 0, 9B + C = 0, C = - 9B.

Fro the other side B = 2π/24 = π/12, because period p = 24 hours. From here C = -9π/12 = -3π/4.

We get finaly equation: T = 5sin(π/12·t - 3π/4) + 80.

Now we have to find first time when T = 79°F.

We get trigonometric equation: 5sin(π/12·t - 3π/4) + 80 = 79, from here sin(π/12·t - 3π/4) = -0.2;

From here π/12·t - 3π/4 =sin^-1(0.2), t =(3π/4- sin^-10.2)/(π/12) = 8.23 hours

At t = 8.23 hoursr AM temperature first time will be 79°F

Answer 2

To create a model from this information, we can draw the following conclusions:

The amplitude is 5 because it is half of the temperature range.

The center line is at y = 80 (also the average temperature) because that is midway between 75 and 85. This is the vertical translation.

The period is 24 (because there are 24 hours in a day), so the multiplier on x is ^2π/₁₂, or ^π/₆.

The sine graph is shifted 9 units to the right because it reaches the average daily temperature for the first time at 9 am.

We can use the general equation y = a sin [b(x - h)] + k, and put in the foregoing information as follows:

y = 5 sin [(^π/₁₂)(x - 9)] + 80

From here, there are a couple of options. I put this into a graphing calculator, along with the equation y = 79. They intersect in two places: one on the way up and one on the way down. I used the solver to get the x value while the temperature is on its way up, and I got 8.23 as my answer.

This can also be done by setting the equation equal to 79, and solving for x. This is not difficult, and it yields the same result.

Answer 3

Just reading the problem, no it is not 9.76

The average temperature is 80°. This occurs at 9 am.

A temperature of 79° would occur before, less than 9 hours after midnight.

What formula did you use?